Punctured neighborhood theorem

Question

Let $X$ be a complex Banach space and let $T$ be a bounded linear operator acting on $X.$ Consider a subspace $M$ (not necessarily closed) such that $T(M)\subset M,$ $T(X)+M=X$ and there exists $\epsilon>0$ such that $T-\lambda I$ is injective and $M\subset (T-\lambda I)(X)$ for all $0<|\lambda|<\epsilon.$ I showed, using the fact that $T(X)+M=X,$ that there exists $\epsilon\geq\epsilon^{'}>0$ such that $(T-\lambda I)(X)+\overline{M}=X,$ where $\overline{M}$ is the closure subset of $M.$ And hence the operator $(T-\lambda I)$ is injective and with a dense image. My question is, is there $\epsilon^{''}>0$ such that $(T-\lambda I)(X)$ is closed?